A yo-yo of mass m rests on the floor (the static friction coefficient with the floor is μ). The inner (shaded) portion of the yo-yo has a radius R1, the two outer disks have radii R2. A string is wrapped around the inner part. Someone pulls on the string at an angle β (see sketch). The "pull" is very gentle, and is carefully increased until the yo-yo starts to roll without slipping. Try it at Home; it's Fun!

For what angles of β will the yo-yo roll to the left and for what angles to the right?

a)Yo-Yo rolls to the left if sinβ<R1R2, and to the right if sinβ>R1R2.

b)Yo-Yo rolls to the left if sinβ>R1R2, and to the right if sinβ<R1R2.

c)Yo-Yo rolls to the left if cosβ<R1R2, and to the right if cosβ>R1R2.

d)Yo-Yo rolls to the left if cosβ>R1R2, and to the right if cosβ<R1R2.

sorry b)

b) is a wrong ansfer!!

To determine the conditions under which the yo-yo rolls to the left or to the right, we need to consider the forces acting on the yo-yo when it starts rolling without slipping.

When the yo-yo starts rolling, it experiences two main forces: the tension in the string and the static friction between the yo-yo and the floor.

Let's analyze the forces in both scenarios, when the yo-yo rolls to the left and when it rolls to the right:

1. Yo-yo rolls to the left:

In this case, the tension in the string acts towards the left, opposing the rolling motion. The static friction between the yo-yo and the floor acts towards the right, providing the necessary force to initiate the rolling motion.

2. Yo-yo rolls to the right:

In this case, the tension in the string acts towards the right, aiding the rolling motion. The static friction between the yo-yo and the floor acts towards the left, opposing the rolling motion.

Now, let's consider the effect of angle β on these forces. The force from the tension in the string can be resolved into its horizontal and vertical components:

Tension vertical component = T × sin(β)
Tension horizontal component = T × cos(β)

To initiate the rolling motion, the static friction must be greater than or equal to the force opposing the rolling motion. Therefore, the following conditions should be met:

1. Yo-yo rolls to the left:
The static friction force (f_friction) must be greater than or equal to the tension horizontal component (T × cos(β)). Hence, cos(β) should be less than or equal to R1/R2.

2. Yo-yo rolls to the right:
The static friction force (f_friction) must be greater than or equal to the tension horizontal component (T × cos(β)). Hence, cos(β) should be greater than or equal to R1/R2.

Therefore, the correct answer is:

d) Yo-Yo rolls to the left if cos(β) > R1/R2, and to the right if cos(β) < R1/R2.

This is part of an 8.01x problem.

Answers should not be given untill the deadline!!

Cheaters will be banned from 8.01x!